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INTRODUCTION
The line heating method is a popular technique used in
shipyards to form ship hull pieces. Even recently, skilled
workers have performed the forming process by their
empirical intuition and so no systematic way to effective
formation has been found. The automation of it can reduce
working time, rework costs, and inferior hull pieces; this is
so attractive to shipbuilders who wish higher productivity
of ships.
The forming process by heating has much complexity
and uncertain factors that are obstacles to the automation.
Of many difficulties, three main problems can be
summarized as follows:
i) How can the heat-induced plate deformation be pre-
dicted under combinations of many heating conditions?
ii) On which regions of the plate should be heated and bent
to minimize forming errors?
iii) How the techniques-i) and ii) can be integrated for
practicality?
This study deals with the problems and proposes a
solution to overcome them. We developed a simulator of
the plate forming process by the line heating in which heat
regions can be found under specified heat conditions;
traveling speed, heat flux and so forth.
In order to predict the deformation of a plate, a thermal
elasto-plastic analysis model proposed by Jang et al [1, 2]
was adopted. The inherent strain produced in a heated
zone can be substituted with the equivalent forces and
moments in some assumptions: a circular disk-spring
model, relevant inherent strain zones and so on. The
complicated phenomena of the heated plate may be sim-
plified and the deformation can be effectively estimated
because the thermal elasto-plastic behavior of the plate is
analyzed only by elastic finite element method. For large
deformation analysis, Newton-Raphson method is used
considering geometrical non-linearity of the deformed
surface. Residual deformation is our main concern. No
residual stresses were considered in the finite element
analysis, therefore, the errors caused by neglecting them
can be more realistically corrected in the next forming step
[3].
In addition, we employed an algorithm to determine
heating paths [4]. Candidate heat paths can be obtained by
grouping extreme principal curvatures on a deflection
difference surface that represent the difference between a
target surface and a fabrication surface. By a convergence
index that indicates the likelihood of the surfaces, one heat
line is selected among the heat paths. The advantage of
this method is believed to be the ability of adjacent heat
lines to correct some uncertainty of a heating like the
residual stress, which is based upon the step-by-step
determination of the heat regions. The two techniques have
proved to be fast and accurate. It is a desirable character-
istic because the plate forming needs a number of the
1
Professor, Seoul National University, Korea,
2
Ph.D. student, Seoul National University, Korea,
3
Assistant manager, Samsung Heavy Industries, Korea
Acquisition of Line Heating Information for Automatic Plate
Forming
Chang Doo Jang 1
, Sung Choon Moon 2
and Dae Eun Ko 3
Currently, the promotion of productivity is a significant topic in shipbuilding. The line heating, used
for forming curved plates, requires many man-hours and the empirical intuition of skilled shipwrights.
Therefore, the automatic forming method is being widely studied for practical adoption in shipyards.
As an essential part of the automatic plate forming system, a new line heating simulator is suggested
to acquire the heat information. In order to predict the heat-induced plate deformation, a thermal
elasto-plastic analysis model is employed, which is based on the inherent strain method with the
nonlinear finite element method. Heating paths are determined by geometric analysis of a target sur-
face and an initial surface. Also, a surface mesh generation method is developed to link the finite
element analysis and the determination of heat regions. The simulator includes modules of the above
preliminary techniques and heat lines can be obtained. Some hull pieces similar to actual plates are
tested. From the results, it is shown that our simulator can give heating information with good con-
vergence from an initial surface to a target surface. With a line heating robot, it can contribute to the
automation of the line heating process.
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Presentation

2. 2
heating lines and the finite element analysis should be
performed at each heating step.
In the analysis of shells such as ship hulls, quadrilat-
eral elements have been known to show good performance
in comparison with triangular elements. A general shell
element called MITC4 [5] is used. For usage of it, the
generated meshes should be only quadrilateral over a three
dimensional surface domain. Since a heat line breadth is,
in general, very small compared with hull piece's scale, the
mesh must have good transitivity from large elements to
small ones. Thus, the paving method suggested by Blacker
et al [6, 7] is modified to satisfy the conditions and divide
the fabrication surface and heat lines into well-shaping
finite elements automatically, at each step. The surface
mesh generation can play a role to link the two elemental
methods mentioned above.
Moreover, all the surfaces are modeled with NURBS
(Non-Uniform Rational B-spline Surface) to be compatible
with CAD systems or ship product models. The surfaces
are used in the differential geometry analysis and the mesh
generation.
PROCEDURE OF THE SIMULATION
The objective of the simulation is to acquire the
heating information necessary to form a given initial
surface to a given target surface; heat regions, heating
order, etc. By the algorithm of heat line determination, one
heat line can be found and meshes are generated according
to the line location and surface geometry. Then finite
element analysis is performed with the equivalent nodal
forces and moments calculated using given heat conditions.
The fabrication surface is updated based on the surface
deformation resulting from the finite element analysis.
Next, convergence to the target surface is checked. The
convergence index represents the resemblance of the target
and deformed surfaces; if it is zero, the surfaces are
perfectly identical. When the surface is converged enough
to the target surface or no more heating lines are deter-
mined, the simulation is terminated. The overall procedure
is illustrated in figure 1.
SIMULATION EXAMPLES
Several surface types were selected, similar to actual
ship hull pieces; saddle type, concave type and skewed
saddle type. Based on experimental data we assumed that
heat conditions were fixed. Table 1 shows that the data and
their equivalent loads calculated using the analysis model.
A heat line is curvilinear and so the forces and moments
FE analysis
Convergence checkEnd Yes
Heat line determination Mesh generation
Update of fabrication plate
Target geometry and initial plate
No
Figure 1 Overall procedure of the simulation of the line heating process
Table 1 Heat Conditions and equivalent forces
Traveling speed 0.3889 cm/sec
Heat input 2250 cal/sec
Inherent strain 0.00639
Heat line breadth 20 mm
Transverse bending moment (my) 220.8 kgf
Transverse shrinkage force (fy) 31.9 kgf/mm
Longitudinal bending moment (Mx) 25534.1 kgf mm
Longitudinal shrinkage force (Fx) 3684.2 kgf

3. 3
should be loaded along the curve, that is, a geodesic
connecting the heat line’s start and end points. Figure 2
describes the distribution of the forces and the force
boundary conditions of each element on the heat line.
Simple line heating
Using the conditions in table 1, we analyzed a simple
heating example to know the behavior of the plate under a
line heating. The heat line crosses a plate with the
dimensions 1000×1000×20 (mm3
). The generated meshes
of the deformed shape are shown in figure 3.
The maximum relative deflection of the plate along
A-A section was 0.301mm and along B-B section was
1.693mm. From the result, the heat line would bend the
flat plate in heating direction as well as the perpendicular
direction, that is, to a concave configuration. This is due to
the longitudinal and transverse moments acting together on
the plate with the shrinkage forces.
In case of the saddle and concave surfaces, we
imposed the different amount of curvature on the initial
deformed surfaces to grasp the effect of the initial cur-
vature. The plates’ dimension is 1500×1000×20(mm3
).
Tables 2, 3 and 4 show the geometry of the simulation
examples and obtained heating lines. In each picture, solid
and dotted lines represent front face heating and back side
heating respectively. To recognize the fabrication errors,
the target surface (gray color) and the deformed surfaces
(black color) are also drawn. The deflection of all the
surfaces is magnified five times for easy view. The heating
lines’ information is described in table 5 and the conver-
Table 2 Geometry of the saddle simulations and obtained results
Surface type
Target surface and
initial surface
Heating lines Target and deformed surfaces
Saddle I
h1 = 12mm
h2 = 15mm
h = 12mm
Saddle II
h1 = 12mm
h2 = 15mm
h = 14mm
h1 h2
h
1500×1000×20 mm3
A
A
B B
Heat line
Heated plate Mesh for FEA (deformed shape)
Figure 3 Simple line heating example
(magnified 30 times)
b
fy
Mx
Fx my
fy
Mx
Fx
Equivalent forces and moments along a heat line
Mx0.5
fy
0.5l fy
0.5l
fy
0.5l fy
0.5l
my0.5l
b
l
Mx0.5
Mx
0.5Mx0.5
Fx0.5
my0.5l
my0.5l
my0.5lFx
0.5
Fx
0.5
Fx
0.5
Load conditions at an element
Figure 2 Equivalent loads

4. 4
gence index during the heating process in figure 4.
Saddle type
For the test of more realistic cases, we first adopted the
saddle surface as the target surface of test simulation. The
initial surfaces were modeled by cylindrical or single bent
plate. To one initial surface, the curvature in the transverse
direction was assumed to be identical to that of the target
surface (we call it Saddle I for convenience), while to the
other the initial curvature was larger than it (Saddle II). As
can be seen in table 2, the trend of the acquired heating
lines are in agreement with those we can guess intuitively.
The front face was heated first until no more candidate
heating lines on the face were found. If a plate didn't
converge on the target, it was turned over and heated. The
convergence index decreases gradually to 0.06, the con-
vergence index objective value, with the exception that it
increases slightly after the first turnover (figure 4a).
The number of the heat lines in Saddle II is smaller
and the total heating length is shorter than those of the
Saddle I. This is because line heating is likely to produce
both transverse and longitudinal bending as mentioned in
Table 3 Geometry of the concave simulations and obtained results
Surface type
Target surface and
initial surface
Heating lines Target and deformed surfaces
Concave I
h1 =12mm
h2 =15mm
h =12mm
Concave II
h1 =12mm
h2 =15mm
h =10mm
h1 h2
h
1500×1000×20 mm3
Table 4 Geometry of the skewed saddle simulations and obtained results
Surface type
Target surface and
initial surface
Heating lines Target and deformed surfaces
Skewed
saddle type
h1 = 30mm
h2 = 20mm
h3 = 20mm
h4 = 15mm
ha = 30mm
hb = 20mm
L = 4000mm
B1 = 2000mm
B2 = 1500mm
h1
h2
h3
h4
ha
hb
L
B1
B2

5. 5
the simple heating example. So the curvature of the initial
surface was reduced in reverse due to the opposite side
heating. When fabricating the saddle type surface, there-
fore, it is effective to bend the initial surface some more
than the intended longitudinal curvature. This seems to be
consistent with the actual work experience in shipyards.
Due to the limits of the laboratory experiment, the heat
flux used here was small compared with the intensity of the
workers’ gas torch. From this reason, the interval of the
obtained heat lines tends to be narrow and irregular. We
are developing an improved method to locate the heating
lines evenly or regularly considering the practical heating
process.
Concave type
The concave type has positive Gaussian curvature
over the entire surface. As in the saddle type simulation,
two initial surfaces were tested. The longitudinal curva-
ture of one is equal to that of the target surface (Concave I)
but a less curvature than it was given to the other (Concave
II).
The heat lines are placed in the longitudinal direction
to bend the plate transversely, as depicted in Table 3. In
case of Concave II, the transverse heat line was first
calculated and would deform the plate in the longitudinal
direction owing to the discrepancy of the longitudinal
deflection between the target and initial surfaces. Thus the
total heating length of the concave II is longer, although the
numbers of the lines are same and we could expect that
Concave II had the better convergence, considering the
plate deformation under a heating. Actually, to fabricate
this type of surface, triangle heating is usually used. In
other words, the transverse curve of the concave type can
be generated by the shrinkage concentrated on the plate’s
edge in the longitudinal direction. Thus we can see that it
is important to reflect the coupling effect of the curvature
and inplane shrinkage in forming the concave surfaces.
Even in the simple cases simulated here, it can be
revealed that the feature of the heat deformation and the
target surface’s geometry should be considered in
determining the curvature of the initial cylindrical surface.
Shipwrights have done it at the mechanical bending stage
by roll pressing prior to the heating process.
The highest fabrication error in the finished plate, as
well as in the saddle types, was within an acceptable range,
1.5mm.
Table 5 Numbers of the heat lines and heating lengths (mm)
Saddle I Saddle II Concave I Concave II
Skewed
saddle
Total heat lines 37 28 11 11 31
Face lines 23 21 11 11 22
Back lines 14 7 0 0 9
Turnovers 2 2 0 0 2
Total heating length 25571.2 21982.7 10806.3 11472.2 41083.1
Face heating length 17657.4 17018.3 10806.3 11472.2 23785.3
Back heating length 7913.9 4964.4 0. 0. 17297.8
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 10 15 20 25 30 35 40
No. heat
Saddle I
Saddle II
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 3 6 9 12
No. heat
Concave I
Concave II
0
0.05
0.1
0.15
0.2
0.25
0 5 10 15 20 25 30
No. heat
(a) Saddle types (b) Concave types (c) Skewed saddle type
Figure 4 Convergence indices

6. 6
Skewed saddle type
In this try, the geometry of the target surface was of
saddle and skewed type and it originated from a real ship
hull piece. The initial surface was singly curved. Heat
lines and the information of heating are demonstrated in
table 4. The maximum error between the target and fab-
ricated plates was 2.2mm and two turnovers were required.
CONCLUSIONS
We present a simulator to acquire heat information to
automate the ship hull bending process. The main infor-
mation is heat regions, heating order, and turnovers with a
given heat flux. It contains the modules of the heat line
determination, prediction of plate deformation by heating,
automatic mesh generation, and other methods for surface
modeling. The heat line allocation patterns were similar to
the yard work. We can know that the plate geometry and
behavior of heated plate should be taken into account when
flat plates are formed initially and mechanically to set up
the heating process generating the compound curvature.
Though not considered here, a series of interface
functions with the heating robot have been implemented
for the actual ship hull manufacturing, and the plate
behavior can be visualized for the worker to check the
heating procedures. More research to enhance the simula-
tor is in progress to generate even and regular heating
patterns while minimizing the heat paths, and the clear
relation of inplane with flexural deformations.
REFERENCES
[1] Jang, C.D., Seo, S.I. and Ko, D.E., A Study on the
Prediction of Deformations of Plates Due to Line
Heating Using a Simplified Thermal Elastoplastic
Analysis, Journal of Ship Production, Vol. 13, No. 1,
1997, pp. 22-27.
[2] Ko, D.E., A Study on the Prediction of Deformations of
Plates due to Line Heating using a Thermal
Elasto-Plastic Analysis Model, Ph.D. thesis, Seoul
National University, 1998 (in Korean).
[3] Ueda, Y., Murakawa, H., Mohamed, R.A., Kamichika,
R., Ishiyama, M., Ogawa, J., Development of
Computer-Aided Process Planning System for Plate
Bending by Line Heating (4th
Report) Decision Making
on Heating Condition, Location, and Direction, Journal
of the Society of Naval Architects of Japan, Vol. 171,
1993, pp.683-695 (in Japanese).
[4] Jang, C.D., Moon, S.C., An Algorithm to Determine
Heating Lines for Plate Forming by Line Heating
Method, Journal of Ship Production, Vol. 14, No. 4,
1998, pp. 238-245.
[5] Dvorkin, E.N. and Bathe, K.J., A formulation of
general shell elements - The use of mixed interpolation
of tensorial components", International Journal for
Numerical Methods in Engineering, Vol. 22, 1986,
pp.697-722.
[6] Blacker, T.D. and Stephenson, M. B., Paving: A New
Approach to Automated Quadrilateral Mesh Generation,
International Journal for Numerical Methods in
Engineering, Vol. 32, 1991, pp. 811-847.
[7] Cass, R.J., Benzley, S.E., Meyers, R. J., and Blacker,
T.D., Generalized 3-D Paving: An Automated
Quadrilateral Surface Mesh Generation Algorithm,
International Journal for Numerical Methods in
Engineering, Vol. 39, 1996, pp. 1475-1489.